| As is well known, it is of great importance to study existence and uniqueness of solution to differential equations. Various techniques have been established to study this kinds of problems, such as, variational method, degree theory, critical point theory, implicit function theory, semi-group of operators etc. In this thesis, we discuss the existence and uniqueness for a kind of nonlinear nonresonant ellipticproblem. By defining a real bilinear symmetric functional and applying the Schauder fixed point theorem, the existence and uniqueness for the weak solution of nonlinear nonresonant elliptic system with hardy potential is proved.In 1969, Lazer and Sanchezt [20] applied the Brouwer fixed point theorem to obtain the existence of periodic solutions to Newtion equationu"(t) +▽G(u(t)) = p(t) = p(t + 2π) (4.1)under some conditions. Here G∈C2(Rn,R), p : R→Rn is a continuous 2πperiodic function. In 1973, Ahmad[1] gave a proof of the existence of 2π-periodic solutions for above equation. Lazer [17] applied a real bilinear symmetric functionalto obtain the existence and uniqueness of 2πperiodic solution to the above equation. In 1977, Kannan and Locker[16] obtained a simpler proof for the existenceand uniqueness for the above problem. With a global inverse function theorem, Brown and Lin[9] showed a different proof for the existence and uniquenessto the problem (4.1). By various technique, they proved the following result.Theorem 1.Assume there exist two n×n real constant symmetric matrices A and B such thatA≤Q(x,y))≤B, (x,y)∈Rn×n, and there exist integers Nk(k = 1,2,…, n) such thatNk2 <ξk≤ηk< (Nk +1)2,whereξ1≤ξ2≤…≤ξn andη1≤η2≤…≤ηn are the eigenvalues of A and B, respectively, then the problem (4.1) has a unique 2πperiodic solution.After that, some new results about the global inverse function theorem appeared, and many authors applied these results to study the existence and uniqueness of 2πperiodic solution to problem (4.1) under more weaker conditions. In [34, 35], Z. H. Shen continued to investigate problem (4.1) and obtained a very-general result on the existence and uniqueness of 2πperiodic solution to problem (4.1). In 1989, Y. Li consider the following two-point boundary value problemu"(t) = f(t, u(t)) u(0) = u(1) = 0.He combined the idea of [17] with degree theory to prove the existence and uniquenessof solution to the above problem. In 1991, Y. Li and H. Z. Wang [24] studied a kind of high order ordinary differential equation as followsu2n(t) +▽G(u(t)) = p(t) = p(t + 2π).They obtained a result on the existence and uniqueness of 2πperiodic solution to the above problem. F. Z. Cong applied the method in [17] to study the existenceand uniqueness of 2πperiodic solutions to the following ordinary differential equations with even and odd order respectively. Li [11] [22] extended a nonvariational form of minimax principle and obtained the existence and uniqueness of 2πperiodic solution to above problem at resonance.In the case of partial differential equation, problems at resonance have been of interest to researchers ever since the famous paper of Landsman and Lazer [18] for second order elliptic operators in bounded domain. Subsequently, many papers on the resonant and nonresonant problems for both ordinary and partial differential equations have been published. The interested readers are refered to see [2, 3, 9, 8, 6, 7, 14, 25, 29, 30, 31] and the reference. On the other hand, in recent years, much attention have been paid to the study for elliptic problems with hardy potential as follows[10, 33, 36].-△u-μu/|x|2=f(x,u). However, most of the researchers tried to obtain the existence of nontrivial solutionsfor above problem under the condition that the nonlinearity has a critical growth. In addition, the pervious authors mainly paid attention on scalar equations,a natural question is that if there are similar results for above problem as the case of ODE. In this thesis, we apply the idea of [17] to show that if the ordinary differential operators are replaced by the elliptic operator with hardy potential L=-△-μ/|x|2, a result on the existence and uniqueness of solution for the above elliptic problem in vector case as in the case of ODE can be obtained. This result fills the gap of this topic.Consider the following elliptic system with Dirichlet boundary value conditionswhereΩ∈Rn(n≥2) is a open and bounded domain with smooth boundary (?)Ω, 0∈Ω,μ> 0, 0≤μ<μ= (N-2/2)2, u = (u1,u2,…,un), f = (f1,f2,…,fn), f∈C1(Rn×Rn,Rn).Letσμdenote the spectrum of the operator L with zero Dirichlet boundary condition. In view of [10, 36],σμ(0≤μ<μ) is discrete, contained in the positive semi-axis and each eigenvalueλi(i = 1,2...,) is isolated and has finite multiplicity, the smallest eigenvalueλ1 being simple andλi→∞as i→∞. Repeating each eigenvalue according to its finite multiplicity if necessary, we have a sequence of eigenvalues {λj}j=1∞, where0 <λ1≤λ2≤λ3≤…≤λj→∞.In addition, there exists an orthonormal basis {(?)j}j=1∞of L2(Ω), normalized by‖(?)j‖0=1, where (?)j∈H01(Ω) is an eigenfunction corresponding toλj. It is easily seen that the sequence {(?)j}j=1∞is also complete and orthonormal in H01(Ω). Moreover,‖(?)i‖12=λi. Moreover,‖(?)i‖12 =λi. As in [?], for allμ∈[0,μ), we endow the Hilbert space Hμwith the scalar productHμ=∫Ω(▽u·▽v-μuv/|x|2)dx,(?)u,v∈Hμand define‖u‖Hμ=(∫Ω(|▽u|2-μu2/|x|2)dx)1/2,(?)u∈Hμ.By Hardy inequality we can easily deduce that the norm‖…‖Hμis equivalent to the usual norm in H01(Ω). Note that the spectrum of elliptic operator Lu = -△u-μu/|x|2 is similar to that of ordinary differential operator, and the framework of space Hμwith inner product and norm defined as above is similar that of the space H01(Ω), we prove that there is a similar result for elliptic system as the ordinary system. We first introduce some related lemmas and define a real bilinear symmetric functional to deal with the linear problem. Then we translate the nonlinear problem into a fixed point problem of a completely continuous operator. By showing the range of this operator is bounded, we apply the Schauder fixed point theorem to obtain the existence of fixed point and thus prove the existence of solutions of the nonlinear problem. The uniqueness can be deduced by the result of linear problem directly.
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